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Laplace and Z Transforms - MIT

: Signals and SystemsLaplace and Z TransformsOctober 1, 2009 Mid-term Examination #1 Wednesday, October 7, 7:30-9:30pm, Walker recitations on the day of the :DT Signals and SystemsLectures 1 5 Homeworks 1 4 Homework 4 will include practice problems for mid-term , it will not collected or graded. Solutions will be book: 1 page of notes (812 11inches; front and back).Designed as 1-hour exam; two hours to sessions during open o ce ict? Contact before Friday, October 2, TimeMany continuous-time systems can be represented with di : leaky tankr0(t)r1(t)h1(t)Di erential equation representation: r1(t) =r0(t) r1(t)Last time we considered two methods to solve di erential equations: solving homogeneous and particular equations singularity matchingSolving Di erential Equations with Laplace TransformThe Laplace transform provides a particularly powerful method ofsolving di erential equations it Transforms a di erential equationinto an algebraic (whereLrepresents the Laplace transform ):di erentialalgebraicalgebraicdi erential equation solve di erentialequationalgebraicequationalgebra icanswersolution

Laplace transform: s2Y(s)+3sY(s)+2Y(s) = 1 Solve: Y(s) = 1 (s+1)(s+2) = 1 s+1 − 1 s+2 Inverse Laplace transform: y(t) = e−t−e−2t u(t) These forward and inverse Laplace transforms are easy if • dierential equation is linear with constant coecients, and • …

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